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## Gidon Eshel

Print publication date: 2011

Print ISBN-13: 9780691128917

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691128917.001.0001

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# Matrix Properties, Fundamental Spaces, Orthogonality

Chapter:
(p.12) Three Matrix Properties, Fundamental Spaces, Orthogonality
Source:
Spatiotemporal Data Analysis
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691128917.003.0003

This chapter provides an introduction to linear algebra. Topics covered include vector spaces, matrix rank, fundamental spaces associated with A ɛ ℝM×N, and Gram–Schmidt orthogonalization. In summary, an M × N matrix is associated with four fundamental spaces. The column space is the set of all M-vectors that are linear combinations of the columns. If the matrix has M independent columns, then the column space is ℝM; otherwise the column space is a subspace of ℝM. Also in ℝM is the left null space, the set of all M-vectors that the matrix’s s transpose maps to the zero N-vector. The row space is the set of all N-vectors that are linear combinations of the rows. If the matrix has N independent rows, then the row space is ℝN; otherwise, the row space is a subspace of ℝN. Also in ℝN is the null space, the set of all N-vectors that the matrix maps to the zero M-vector.

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